REAL NUMBERS

1. REAL NUMBERS

2. Euclid’s Division Lemma • Let a and b be any two positive integers.Then there exist unique integers q and r such that a=bq+r, 0rb. • The Euclid’s Division Lemma is a restatement of the long division process and the integers q and r are called the quotient and remainder respectively.

3. Application of the Euclids Division Lemma for H.C.F • Let 2 numbers be a and b. • Apply Euclids Division Lemma to a and b and obtain two whole numbers q1 and r1 such that a = bq1+r1. 0r1<b.

4. Application of the Euclids Division Lemma for H.C.F • If r1= 0, b is the H.C.F of a and b. • If r10 apply Euclidsdivision lemma to b and r1 and obtain 2 whole numbers q1 and r2 such that b = q2r1+r2

5. Application of the Euclids Division Lemma for H.C.F • If r2=0 then r1 is the H.C.F of a and b. • If r20 then apply euclids division lemma to r1 and r2 and continue the steps till the remainder is zero. • The divisor at this stage is the H.C.F of a and b

6. Lets Practice-1 Find HCF OF 455 and 42 by Euclid’s division algorithm 455= 42 x 10 +35 42 = 35 x 1 + 7 35 = 7 x 5 + 0 Since the Remainder has become 0, and we cannot proceed further, therefore the HCF of given numbers is the divisor in last step or the remainder of second last step i.e7 (Answer)

7. Lets Practice-2 • Using Euclids Division Lemma find H.C.F of 180670 and 2937.

8. By Euclids division lemma we have 180670= 2937 X 61 + 1513 2397 = 1513 X 1 + 1424 1513 = 1424 X 1 + 89 1424 = 89 X 16 + 0 HCF(180670, 2937) = HCF (2937, 1513) = HCF (1513, 1424) = HCF ( 1424, 89) = 89 Hence HCF(180670, 2937) = 89

9. Class Assignment Using Euclid’s Division Algorithm find HCF of 420 and 130 12576 and 4052 Home Assignment Ex: 1.1 Question 1

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